Volumetric scattered data modeling based on modified Shepard's method

Abstract

Scattered data is defined as a collection of data that have little specified connectivity among data points. Trivariate scattered data interpolation from R3 --> R consists of constructing a function f = (x, y, z) such that f(xi, yi, zi) = Fi, i=1, N where V = {vi = (xi, yi, zi) e R3, i=1,....N} is a set of distinct and non-coplanar data points and F = (F1, ......, FN) is a real data vector. The weighted alpha shapes method is defined for a finite set of weighted points. Let S s Rd x R be such a set. A weighted point is denoted as p=(p', o) with p' e Rd its location and o e R its weight. For a weighted point p and a real a define P+a=(P', o + a). So p and P+a share the same location and their weights differ by a. In other words, it is a polytope uniquely determined by the points, their weights, and a parameter a e R that controls the desired level of detail. Therefore, how to assign the weight for each point is one of the main tasks to achieve the desirable volumetric scattered data interpolation. In other words, we need to investigate the way to achieve different levels of detail in a single shape by assigning weights to the data points. In this paper, Modified Shepard's method is applied in terms of least squares manner.